Motzkin path decompositions of functionals in noncommutative probability

Abstract

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional [Formula: see text] are naturally described in terms of Motzkin paths identified with reduced Motzkin words [Formula: see text]. Using this fact, we demonstrate that the mixed moments of order [Formula: see text] of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to [Formula: see text] with summation extending over [Formula: see text], the set of reduced Motzkin paths of length [Formula: see text]. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice [Formula: see text] into sublattices [Formula: see text] of partitions which are monotonically adapted to colors in [Formula: see text]. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.